The graph of $f$

Question

My question is kind of silly but I’m wondering why in my textbook, their is a drawing of a function $f$ whose graph continues after the limit point into the $x_0 +\delta$ region of the interval. I don’t understand why this is so. I think about $\frac{1}{x}$ whose graph ends at $L=0$ not having a graph that extends into the $x_0 +\delta$ region. I would appreciate some help with this small misunderstanding.

score 1 · Answer 1 · answered Jun 07 '21 at 07:01

1

If a function is defined on let say an interval $(a,b)$ and that $x_0 \in (a,b)$, $f$ may have a limit $\lim\limits_{x \to x_0} f(x)=L$ which enables to draw the graph of $f$ on the left and right side of $x_0$. This will be the case by the way for any $x_0 \in (a,b)$ if $f$ is supposed to be continuous on $(a,b)$.

answered Jun 07 '21 at 07:01

mathcounterexamples.net

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What if I choose $x_0=b$ then shouldnt there only be a graph to the left of $x_0$? – Fredsaffiyre Jun 07 '21 at 07:08
@Fredsaffiyre For sure! – mathcounterexamples.net Jun 07 '21 at 07:09
And isnt it was necessary that $x_0=L$ in the definition. If this is the case shouldn’t the graph always be to the left of $x_0$? – Fredsaffiyre Jun 07 '21 at 07:10
$L$ is in the codomain of $f$. So there is absolutely no reason for having $x_0=L$. – mathcounterexamples.net Jun 07 '21 at 07:26

The graph of $f$

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